This function computes \(d_{Mod}\) effect sizes from user-defined descriptive statistics
and regression coefficients. If one has access to a raw data set, the `dMod`

function may be used
as a wrapper to this function so that the regression equations and descriptive statistics can
be computed automatically within the program.

```
compute_dmod_par(referent_int, referent_slope, focal_int, focal_slope,
focal_mean_x, focal_sd_x, referent_sd_y, focal_min_x, focal_max_x,
focal_names = NULL, rescale_cdf = TRUE)
```

referent_int

Referent group's intercept.

referent_slope

Referent group's slope.

focal_int

Focal groups' intercepts.

focal_slope

Focal groups' slopes.

focal_mean_x

Focal groups' predictor-score means.

focal_sd_x

Focal groups' predictor-score standard deviations.

referent_sd_y

Referent group's criterion standard deviation.

focal_min_x

Focal groups' minimum predictor scores.

focal_max_x

Focal groups' maximum predictor scores.

focal_names

Focal-group names. If `NULL`

(the default), the focal groups will be given numeric labels ranging from 1 through the number of groups.

rescale_cdf

Logical argument that indicates whether parametric \(d_{Mod}\) results
should be rescaled to account for using a cumulative density < 1 in the computations (`TRUE`

; default) or not (`FALSE`

).

A matrix of effect sizes (\(d_{Mod_{Signed}}\),
\(d_{Mod_{Unsigned}}\), \(d_{Mod_{Under}}\),
\(d_{Mod_{Over}}\)), proportions of under- and over-predicted criterion scores,
minimum and maximum differences (i.e., \(d_{Mod_{Under}}\) and \(d_{Mod_{Over}}\)),
and the scores associated with minimum and maximum differences.
Note that if the regression lines are parallel and infinite `focal_min_x`

and `focal_max_x`

values were
specified, the extrema will be defined using the scores 3 focal-group SDs above and below the corresponding focal-group means.

The \(d_{Mod_{Signed}}\) effect size (i.e., the average of differences in prediction over the range of predictor scores) is computed as $$d_{Mod_{Signed}}=\frac{1}{SD_{Y_{1}}}\intop f_{2}(X)\left[X\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right] dX,$$ where

\(SD_{Y_{1}}\) is the referent group's criterion standard deviation;

\(f_{2}(X)\) is the normal-density function for the distribution of focal-group predictor scores;

\(b_{1_{1}}\) and \(b_{1_{0}}\) are the slopes of the regression of \(Y\) on \(X\) for the referent and focal groups, respectively;

\(b_{0_{1}}\) and \(b_{0_{0}}\) are the intercepts of the regression of \(Y\) on \(X\) for the referent and focal groups, respectively; and

the integral spans all \(X\) scores within the operational range of predictor scores for the focal group.

The \(d_{Mod_{Under}}\) and \(d_{Mod_{Over}}\) effect sizes are computed using the same equation as \(d_{Mod_{Signed}}\), but \(d_{Mod_{Under}}\) is the weighted average of all scores in the area of underprediction (i.e., the differences in prediction with negative signs) and \(d_{Mod_{Over}}\) is the weighted average of all scores in the area of overprediction (i.e., the differences in prediction with negative signs).

The \(d_{Mod_{Unsigned}}\) effect size (i.e., the average of absolute differences in prediction over the range of predictor scores) is computed as $$d_{Mod_{Unsigned}}=\frac{1}{SD_{Y_{1}}}\intop f_{2}(X)\left|X\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right|dX.$$

The \(d_{Min}\) effect size (i.e., the smallest absolute difference in prediction observed over the range of predictor scores) is computed as $$d_{Min}=\frac{1}{SD_{Y_{1}}}Min\left[\left|X\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right|\right].$$

The \(d_{Max}\) effect size (i.e., the largest absolute difference in prediction observed over the
range of predictor scores)is computed as
$$d_{Max}=\frac{1}{SD_{Y_{1}}}Max\left[\left|X\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right|\right].$$
*Note*: When \(d_{Min}\) and \(d_{Max}\) are computed in this package, the output will display the
signs of the differences (rather than the absolute values of the differences) to aid in interpretation.

If \(d_{Mod}\) effect sizes are to be rescaled to compensate for a cumulative density less than 1 (see the `rescale_cdf`

argument), the result of each
effect size involving integration will be divided by the ratio of the cumulative density of the observed range of scores (i.e., the range bounded by the `focal_min_x`

and `focal_max_x`

arguments) to the cumulative density of scores bounded by `-Inf`

and `Inf`

.

Nye, C. D., & Sackett, P. R. (2016).
New effect sizes for tests of categorical moderation and differential prediction.
*Organizational Research Methods*, https://doi.org/10.1177/1094428116644505.

```
# NOT RUN {
compute_dmod_par(referent_int = -.05, referent_slope = .5,
focal_int = c(.05, 0, -.05), focal_slope = c(.5, .3, .3),
focal_mean_x = c(-.5, 0, -.5), focal_sd_x = rep(1, 3),
referent_sd_y = 1,
focal_min_x = rep(-Inf, 3), focal_max_x = rep(Inf, 3),
focal_names = NULL, rescale_cdf = TRUE)
# }
```

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